The hot water tank should work on gas, even with shore power. Point of fact for those inclined: Water heater recovery can be improved by running both electric and propane heating sources simultaneously. With two "sources of heat", water heats more quickly than with one. Not twice as fast, but, faster
When you flip the water heater switch to "On" (the inside gas switch), the red light should light up until the DSI unit sparks the burner tube to "lit". Since you aren't getting the red light, even momentarily, it would indicate "no power" to the control board/switch. First thing to look at is the fuse box...should be a fuse for the water heater listed.
If that has a good fuse in place, then it starts getting more complicated. You can't light the burner manually if the igniter/igniter circuit isn't functioning...that is a safety control to keep your water heater from spewing raw LP all over the place causing an explosion hazard. Trouble shooting why the board doesn't have power will take some rooting around and some "test tools"; ie., a VOM or at the very least a 12v test light and a pretty good understanding of the wiring involved. Not really something that can be described in a Forum. The control board is mounted near the water heater (usually right next to it) and the water heater has a wiring diagram glued to it...if you are "comfortable" with wiring diagrams and have the ability to read them, they are useful, but, most people don't have the skills to do much with them and the need for a good RV repairman is usually required. But... Before calling one, one thing anyone can do is...follow all the wires. Any place you find a junction, be it a screw post, slip on, etc. examine the connection and do a "wiggle test". If you find a wire nut (the little cone shaped connectors), undo them by twisting them off, check the wires to make sure they are twisted together and screw the wire nut back on. You don't know how many times I've found a loose wire or bad wire nut connection to be source of problem.